Schwarzschild-AdS Lambdavacuum
PDEagle said:
Where is the horizon of the AdS space?
What's the relationship between AdS space and black hole?
Is there any related articles?
thanks.
A good source for basic information about the global structure of the Schwarzschild-dS and Schwarzschild-AdS solutions (and their electrically charged generalizations) is
http://www.arxiv.org/abs/gr-qc/9507019
Note that there are many distinct coordinate charts in which these solutions may be written, which are useful in different contexts, and in addition the metric functions can be parameterized in different ways. In particular, some authors prefer to write them in terms of \Lambda in which case the same line element can be used for either dS or AdS. Others prefer to write a form in which the approximation location of the cosmological horizon is given by one parameter.
An interesting pedagogical point here is that we locate the event and cosmological horizons by solving a polynomial, which in this case happens to be a cubic, e.g. in Schwarschild-dS it can be written 1-2m/r-r^2/a^2. This is a good opportunity to put a solid mathematical education to use by employing the discriminant to ensure that we have three real roots (we only care about the two positive real roots, of course), then using Sturm's method to analyze the disposition of the roots, and finally using multivariable Taylor series to approximation the roots. In the Schwarzschild-dS case, we find r \approx 2m, \; r \approx a-m for the location of the event horizon and cosmological horizon, where a^2 > 27 m^2 and where \Lambda=3/a^2.
A very good exercise for gtr students is to follow the model of the analysis via effective potentials of test particle orbits for the Schwarzschild vacuum to study the orbits of test particles in the Schwarzschild-dS or AdS lambdavacuums. In the dS case, for some values of orbital angular momentum (of the test particle), we have two unstable and one stable circular orbits. More interestingly, test particles with zero orbital angular momentum can "hover" at just the right radius to balance the gravitational attraction of the massive object at the "center" with the effect of positive \Lambda (This configuration is unstable.)
Your second question is a bit too vague for me to attempt to answer. Can you provide some context? What level of mathematical sophistication did you desire in a reply?
Chris Hillman
